Exam 4: Exponential and Logarithmic Functions

Use the Definition of a Logarithm to Solve Logarithmic Equations Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $3 \ln ( 6 x ) = 6$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\log _ { 3 } 135 - \log _ { 3 } 5$

Solve the problem. -The logistic growth function $\mathrm { f } ( \mathrm { t } ) = \frac { 56,000 } { 1 + 1865.7 \mathrm { e } ^ { - 1.9 \mathrm { t } } }$ models the number of people who have become ill with a particular infection t weeks after its initial outbreak in a particular community. How many people were ill after 6 weeks?

Graph the function. -Use the graph of $\log _ { 2 } x$ to obtain the graph of $f ( x ) = 1 + \log _ { 2 } x$ .

Write the equation in its equivalent exponential form. - $\log _ { b } 8 = 3$

Evaluate or simplify the expression without using a calculator. - $10 \log 6$

Graph the function by making a table of coordinates. - $f(x)=\left(\frac{2}{3}\right)^{x}$

Use the Definition of a Logarithm to Solve Logarithmic Equations Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 4 } x + \log _ { 4 } ( x - 15 ) = 2$

Write the equation in its equivalent logarithmic form. - $13 ^ { x } = 169$

Solve the problem. -The function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.0099 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 400 pounds of the material are initially put into the vault, how many pounds will be left after 40 years?

Solve the problem. -The long jump record, in feet, at a particular school can be modeled by f(x)= 19.2 + 2.2 ln (x + 1)where x is the number of years since records began to be kept at the school. What is the record for the long jump 20 years after record started being kept? Round your answer to the nearest tenth.

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $6 \log _ { b } q - \log _ { b } r$

Use Natural Logarithms Evaluate or simplify the expression without using a calculator. - $\ln \mathrm { e } ^ { 11 }$

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $\mathrm { e } ^ { \mathrm { x } } = 3.9$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 5 } \left( \frac { \sqrt { x } } { 125 } \right)$

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $\mathrm { e } ^ { 5 \mathrm { x } - 8 } - 10 = 1178$

Solve the problem. -A sample of $300 \mathrm {~g}$ of lead-210 decays to polonium-210 according to the function given by $\mathrm { A } ( \mathrm { t } ) = 300 \mathrm { e } ^ { - 0.032 \mathrm { t } }$ , where $\mathrm { t }$ is time in years. What is the amount of the sample after 60 years (to the nearest g)?

Find the domain of the logarithmic function. - $f ( x ) = \log _ { 5 } ( x - 9 ) ^ { 2 }$

Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm, and then round to three decimal places. - $y = 400 ( 3.8 ) ^ { x }$

Graph the function. -Use the graph of $f ( x ) = e ^ { x }$ to obtain the graph of $g ( x ) = e ^ { x - 5 }$